MerelyVerbalDisputes andCoordinatingonLogical Constants Greg - - PowerPoint PPT Presentation
MerelyVerbalDisputes andCoordinatingonLogical Constants Greg Restall aap conference july 2015 My Plan Background A Definition A Method and its Cost Preservation Examples The Upshot GregRestall
Greg Restall
aap conference · july 2015
My Plan
Background A Definition A Method … … and its Cost Preservation Examples The Upshot
GregRestall http://consequently.org/presentation/2015/verbal-disputes-aap/ 2 of 61
Why I'm interested in MerelyVerbalDisagreement
I’m interested in disagreement… …and I’m interested in words, and what they mean.
GregRestall http://consequently.org/presentation/2015/verbal-disputes-aap/ 4 of 61
Why I'm interested in MerelyVerbalDisagreement
I’m interested in disagreement… …and I’m interested in words, and what they mean.
GregRestall http://consequently.org/presentation/2015/verbal-disputes-aap/ 4 of 61
Why I'm interested in the topic
In particular, I’m interested in the role that logic and logical concepts might play in clarifying and managing disagreement.
GregRestall http://consequently.org/presentation/2015/verbal-disputes-aap/ 5 of 61
Particular Issues
▶ Disagreement between rival accounts of logic
Monism and Pluralism about logic Ontological relativity ( ) The status of modal vocabulary ( )
GregRestall http://consequently.org/presentation/2015/verbal-disputes-aap/ 6 of 61
Particular Issues
▶ Disagreement between rival accounts of logic ▶ Monism and Pluralism about logic
Ontological relativity ( ) The status of modal vocabulary ( )
GregRestall http://consequently.org/presentation/2015/verbal-disputes-aap/ 6 of 61
Particular Issues
▶ Disagreement between rival accounts of logic ▶ Monism and Pluralism about logic ▶ Ontological relativity (∃)
The status of modal vocabulary ( )
GregRestall http://consequently.org/presentation/2015/verbal-disputes-aap/ 6 of 61
Particular Issues
▶ Disagreement between rival accounts of logic ▶ Monism and Pluralism about logic ▶ Ontological relativity (∃) ▶ The status of modal vocabulary (♢)
GregRestall http://consequently.org/presentation/2015/verbal-disputes-aap/ 6 of 61
William James, a Tree, a Squirrel and a Man
A man walks rapidly around a tree, while a squirrel moves on the tree
Does the man go round the squirrel or not? : The man goes round the squirrel. : The man doesn’t go round the squirrel.
GregRestall http://consequently.org/presentation/2015/verbal-disputes-aap/ 8 of 61
William James, a Tree, a Squirrel and a Man
A man walks rapidly around a tree, while a squirrel moves on the tree
Does the man go round the squirrel or not? : The man goes round the squirrel. : The man doesn’t go round the squirrel.
GregRestall http://consequently.org/presentation/2015/verbal-disputes-aap/ 8 of 61
William James, a Tree, a Squirrel and a Man
A man walks rapidly around a tree, while a squirrel moves on the tree
Does the man go round the squirrel or not? α: The man goes round the squirrel. δ: The man doesn’t go round the squirrel.
GregRestall http://consequently.org/presentation/2015/verbal-disputes-aap/ 8 of 61
William James, a Tree, a Squirrel and a Man
Which party is right depends on what you practically mean by ‘going round’ the squirrel. If you mean passing from the north of him to the east, then to the south, then to the west, and then to the north of him again, obviously the man does go round him, for he occupies these successive positions. But if on the contrary you mean being first in front of him, then on the right of him then behind him, then on his left, and finally in front again, it is quite as obvious that the man fails to go round him … Make the distinction, and there is no occasion for any farther dispute. — William James, Pragmatism (1907)
GregRestall http://consequently.org/presentation/2015/verbal-disputes-aap/ 9 of 61
Resolving a dispute by clarifying meanings α: The man goes round1 the squirrel. δ: The man doesn’t go round2 the squirrel. Once we disambiguate “going round” no disagreement remains.
GregRestall http://consequently.org/presentation/2015/verbal-disputes-aap/ 10 of 61
Resolving a dispute by clarifying meanings α: The man goes round1 the squirrel. δ: The man doesn’t go round2 the squirrel. Once we disambiguate “going round” no disagreement remains.
GregRestall http://consequently.org/presentation/2015/verbal-disputes-aap/ 10 of 61
Resolution by translation
▶ For James, “going round1” and “going round2” are
explicated in other terms of α and δ’s vocabulary. Perhaps terms and can’t be explicated in terms of prior vocabulary. No matter. could learn while could learn .
GregRestall http://consequently.org/presentation/2015/verbal-disputes-aap/ 11 of 61
Resolution by translation
▶ For James, “going round1” and “going round2” are
explicated in other terms of α and δ’s vocabulary.
▶ Perhaps terms t1 and t2 can’t be explicated in terms of
prior vocabulary. No matter. could learn while could learn .
GregRestall http://consequently.org/presentation/2015/verbal-disputes-aap/ 11 of 61
Resolution by translation
▶ For James, “going round1” and “going round2” are
explicated in other terms of α and δ’s vocabulary.
▶ Perhaps terms t1 and t2 can’t be explicated in terms of
prior vocabulary. No matter.
▶ α could learn t2 while δ could learn t1.
GregRestall http://consequently.org/presentation/2015/verbal-disputes-aap/ 11 of 61
Introducing General Scheme
A
Lα
A
Lδ
GregRestall http://consequently.org/presentation/2015/verbal-disputes-aap/ 12 of 61
Introducing General Scheme
A
Lα
A
Lδ
tα(A) tδ(A)
L∗
tα tδ
GregRestall http://consequently.org/presentation/2015/verbal-disputes-aap/ 12 of 61
What is a Language? A syntax positions , where each member of is asserted and each member of is denied, which are either incoherent (out of bounds) ,
. + identity: . + weakening: If then and . + cut: If and then .
GregRestall http://consequently.org/presentation/2015/verbal-disputes-aap/ 13 of 61
What is a Language?
▶ A syntax
positions , where each member of is asserted and each member of is denied, which are either incoherent (out of bounds) ,
. + identity: . + weakening: If then and . + cut: If and then .
GregRestall http://consequently.org/presentation/2015/verbal-disputes-aap/ 13 of 61
What is a Language?
▶ A syntax ▶ positions [X : Y], where each member of X is asserted
and each member of Y is denied, which are either incoherent (out of bounds) ,
. + identity: . + weakening: If then and . + cut: If and then .
GregRestall http://consequently.org/presentation/2015/verbal-disputes-aap/ 13 of 61
What is a Language?
▶ A syntax ▶ positions [X : Y], where each member of X is asserted
and each member of Y is denied, which are either incoherent (out of bounds) X ⊢ Y,
. + identity: . + weakening: If then and . + cut: If and then .
GregRestall http://consequently.org/presentation/2015/verbal-disputes-aap/ 13 of 61
What is a Language?
▶ A syntax ▶ positions [X : Y], where each member of X is asserted
and each member of Y is denied, which are either incoherent (out of bounds) X ⊢ Y,
+ identity: . + weakening: If then and . + cut: If and then .
GregRestall http://consequently.org/presentation/2015/verbal-disputes-aap/ 13 of 61
What is a Language?
▶ A syntax ▶ positions [X : Y], where each member of X is asserted
and each member of Y is denied, which are either incoherent (out of bounds) X ⊢ Y,
+ identity: A ⊢ A. + weakening: If then and . + cut: If and then .
GregRestall http://consequently.org/presentation/2015/verbal-disputes-aap/ 13 of 61
What is a Language?
▶ A syntax ▶ positions [X : Y], where each member of X is asserted
and each member of Y is denied, which are either incoherent (out of bounds) X ⊢ Y,
+ identity: A ⊢ A. + weakening: If X ⊢ Y then X, A ⊢ Y and X ⊢ A, Y. + cut: If and then .
GregRestall http://consequently.org/presentation/2015/verbal-disputes-aap/ 13 of 61
What is a Language?
▶ A syntax ▶ positions [X : Y], where each member of X is asserted
and each member of Y is denied, which are either incoherent (out of bounds) X ⊢ Y,
+ identity: A ⊢ A. + weakening: If X ⊢ Y then X, A ⊢ Y and X ⊢ A, Y. + cut: If X ⊢ A, Y and X, A ⊢ Y then X ⊢ Y.
GregRestall http://consequently.org/presentation/2015/verbal-disputes-aap/ 13 of 61
What is a Translation?
may be incoherence preserving: . may be coherence preserving: . may be compositional (e.g., , so .)
GregRestall http://consequently.org/presentation/2015/verbal-disputes-aap/ 14 of 61
What is a Translation?
t : L1 → L2
may be incoherence preserving: . may be coherence preserving: . may be compositional (e.g., , so .)
GregRestall http://consequently.org/presentation/2015/verbal-disputes-aap/ 14 of 61
What is a Translation?
t : L1 → L2
▶ t may be incoherence preserving: X ⊢L1 Y ⇒ t(X) ⊢L2 t(Y).
may be coherence preserving: . may be compositional (e.g., , so .)
GregRestall http://consequently.org/presentation/2015/verbal-disputes-aap/ 14 of 61
What is a Translation?
t : L1 → L2
▶ t may be incoherence preserving: X ⊢L1 Y ⇒ t(X) ⊢L2 t(Y). ▶ t may be coherence preserving: X ̸⊢L1 Y ⇒ t(X) ̸⊢L2 t(Y).
may be compositional (e.g., , so .)
GregRestall http://consequently.org/presentation/2015/verbal-disputes-aap/ 14 of 61
What is a Translation?
t : L1 → L2
▶ t may be incoherence preserving: X ⊢L1 Y ⇒ t(X) ⊢L2 t(Y). ▶ t may be coherence preserving: X ̸⊢L1 Y ⇒ t(X) ̸⊢L2 t(Y). ▶ t may be compositional (e.g., t(A ∧ B) = ¬(¬t(A) ∨ ¬t(A)), so
t(λp.λq.(p ∧ q)) = λp.λq.(¬(¬p ∨ ¬q)).)
GregRestall http://consequently.org/presentation/2015/verbal-disputes-aap/ 14 of 61
Example Translations
▶ tα(going round) = going round1; tδ(going round) = going round2.
, a de Morgan translation. . This is coherence and incoherence preserving, and compositional. , interpreting arithmetic into set theory.
This is compositional and coherence preserving, but not incoherence preserving for fol derivability. is true in all models (whether the axioms of pa hold or not). Its translation is a zf theorem but not true in all models. while .
GregRestall http://consequently.org/presentation/2015/verbal-disputes-aap/ 15 of 61
Example Translations
▶ tα(going round) = going round1; tδ(going round) = going round2. ▶ dm : L[∧, ∨, ¬] → L[∨, ¬], a de Morgan translation.
dm(A ∧ B) = ¬(¬dm(A) ∨ ¬dm(B)). This is coherence and incoherence preserving, and compositional. , interpreting arithmetic into set theory.
This is compositional and coherence preserving, but not incoherence preserving for fol derivability. is true in all models (whether the axioms of pa hold or not). Its translation is a zf theorem but not true in all models. while .
GregRestall http://consequently.org/presentation/2015/verbal-disputes-aap/ 15 of 61
Example Translations
▶ tα(going round) = going round1; tδ(going round) = going round2. ▶ dm : L[∧, ∨, ¬] → L[∨, ¬], a de Morgan translation.
dm(A ∧ B) = ¬(¬dm(A) ∨ ¬dm(B)). This is coherence and incoherence preserving, and compositional.
▶ s : L[0,′ , +, ×] → L[∈], interpreting arithmetic into set theory.
This is compositional and coherence preserving, but not incoherence preserving for fol derivability. is true in all models (whether the axioms of pa hold or not). Its translation is a zf theorem but not true in all models. while .
GregRestall http://consequently.org/presentation/2015/verbal-disputes-aap/ 15 of 61
Example Translations
▶ tα(going round) = going round1; tδ(going round) = going round2. ▶ dm : L[∧, ∨, ¬] → L[∨, ¬], a de Morgan translation.
dm(A ∧ B) = ¬(¬dm(A) ∨ ¬dm(B)). This is coherence and incoherence preserving, and compositional.
▶ s : L[0,′ , +, ×] → L[∈], interpreting arithmetic into set theory.
This is compositional and coherence preserving, but not incoherence preserving for fol
hold or not). Its translation (∀x ∈ ω)(∃y ∈ ω)(∀z)(z ∈ y ≡ (z ∈ x ∨ z = x)) is a zf theorem but not true in all models. while .
GregRestall http://consequently.org/presentation/2015/verbal-disputes-aap/ 15 of 61
Example Translations
▶ tα(going round) = going round1; tδ(going round) = going round2. ▶ dm : L[∧, ∨, ¬] → L[∨, ¬], a de Morgan translation.
dm(A ∧ B) = ¬(¬dm(A) ∨ ¬dm(B)). This is coherence and incoherence preserving, and compositional.
▶ s : L[0,′ , +, ×] → L[∈], interpreting arithmetic into set theory.
This is compositional and coherence preserving, but not incoherence preserving for fol
hold or not). Its translation (∀x ∈ ω)(∃y ∈ ω)(∀z)(z ∈ y ≡ (z ∈ x ∨ z = x)) is a zf theorem but not true in all models. ⊢ (∀x)(∃y)(y = x + 1) while ̸⊢ t[(∀x)(∃y)(y = x + 1)].
GregRestall http://consequently.org/presentation/2015/verbal-disputes-aap/ 15 of 61
A General Scheme… A dispute between a speaker
, and
language , over (where asserts and denies ) is said to be resolved by translations and iff For some language , , and , and .
GregRestall http://consequently.org/presentation/2015/verbal-disputes-aap/ 16 of 61
A General Scheme… A dispute between a speaker α of language Lα, and
language , over (where asserts and denies ) is said to be resolved by translations and iff For some language , , and , and .
GregRestall http://consequently.org/presentation/2015/verbal-disputes-aap/ 16 of 61
A General Scheme… A dispute between a speaker α of language Lα, and δ of language Lδ,
(where asserts and denies ) is said to be resolved by translations and iff For some language , , and , and .
GregRestall http://consequently.org/presentation/2015/verbal-disputes-aap/ 16 of 61
A General Scheme… A dispute between a speaker α of language Lα, and δ of language Lδ, over C (where asserts and denies ) is said to be resolved by translations and iff For some language , , and , and .
GregRestall http://consequently.org/presentation/2015/verbal-disputes-aap/ 16 of 61
A General Scheme… A dispute between a speaker α of language Lα, and δ of language Lδ, over C (where α asserts C and δ denies C) is said to be resolved by translations and iff For some language , , and , and .
GregRestall http://consequently.org/presentation/2015/verbal-disputes-aap/ 16 of 61
A General Scheme… A dispute between a speaker α of language Lα, and δ of language Lδ, over C (where α asserts C and δ denies C) is said to be resolved by translations tα and tδ iff For some language , , and , and .
GregRestall http://consequently.org/presentation/2015/verbal-disputes-aap/ 16 of 61
A General Scheme… A dispute between a speaker α of language Lα, and δ of language Lδ, over C (where α asserts C and δ denies C) is said to be resolved by translations tα and tδ iff
▶ For some language L∗, tα : Lα → L∗, and tδ : Lδ → L∗,
and .
GregRestall http://consequently.org/presentation/2015/verbal-disputes-aap/ 16 of 61
A General Scheme… A dispute between a speaker α of language Lα, and δ of language Lδ, over C (where α asserts C and δ denies C) is said to be resolved by translations tα and tδ iff
▶ For some language L∗, tα : Lα → L∗, and tδ : Lδ → L∗, ▶ and tα(C) ̸⊢L∗ tδ(C).
GregRestall http://consequently.org/presentation/2015/verbal-disputes-aap/ 16 of 61
…and its Upshot Given a resolution by translation, there is no disagreement over C in the shared language L∗. The position (in ) is coherent.
GregRestall http://consequently.org/presentation/2015/verbal-disputes-aap/ 17 of 61
…and its Upshot Given a resolution by translation, there is no disagreement over C in the shared language L∗. The position [tα(C) : tδ(C)] (in L∗) is coherent.
GregRestall http://consequently.org/presentation/2015/verbal-disputes-aap/ 17 of 61
Taking Disputes to be Resolved by Translation To take a dispute to be resolved by translation is to take there to be a pair of translations that resolves the dispute. (You may not even have the translations in hand.)
GregRestall http://consequently.org/presentation/2015/verbal-disputes-aap/ 18 of 61
Taking Disputes to be Resolved by Translation To take a dispute to be resolved by translation is to take there to be a pair of translations that resolves the dispute. (You may not even have the translations in hand.)
GregRestall http://consequently.org/presentation/2015/verbal-disputes-aap/ 18 of 61
… to resolve any dispute by translation.
GregRestall http://consequently.org/presentation/2015/verbal-disputes-aap/ 20 of 61
Resolution by DisjointUnion
GregRestall http://consequently.org/presentation/2015/verbal-disputes-aap/ 21 of 61
Resolution by DisjointUnion
Or, what I like to call “the way of the undergraduate relativist.”
GregRestall http://consequently.org/presentation/2015/verbal-disputes-aap/ 21 of 61
Resolution by DisjointUnion
C
Lα
C
Lδ
tα(C)
Lα|δ = Lα ⊔ Lδ
tδ(C)
Lα|δ = Lα ⊔ Lδ
tα tδ
GregRestall http://consequently.org/presentation/2015/verbal-disputes-aap/ 21 of 61
Resolution by DisjointUnion
C
Lα
C
Lδ
C
Lα|δ = Lα ⊔ Lδ
C
Lα|δ = Lα ⊔ Lδ
tα tδ
GregRestall http://consequently.org/presentation/2015/verbal-disputes-aap/ 21 of 61
Resolution by DisjointUnion
Lα|δ is the disjoint union Lα ⊔ Lδ, and tα : Lα → Lα|δ, tδ : Lδ → Lα|δ are the obvious injections. For coherence on , iff
. This is a coherence relation. The vocabularies slide past one another with no interaction. This ‘translation’ is structure preserving, and coherence and incoherence preserving too.
GregRestall http://consequently.org/presentation/2015/verbal-disputes-aap/ 22 of 61
Resolution by DisjointUnion
Lα|δ is the disjoint union Lα ⊔ Lδ, and tα : Lα → Lα|δ, tδ : Lδ → Lα|δ are the obvious injections. For coherence on Lα|δ, (Xα, Xδ ⊢ Yα, Yδ) iff (Xα ⊢ Yα) or (Xδ ⊢ Yδ). This is a coherence relation. The vocabularies slide past one another with no interaction. This ‘translation’ is structure preserving, and coherence and incoherence preserving too.
GregRestall http://consequently.org/presentation/2015/verbal-disputes-aap/ 22 of 61
Resolution by DisjointUnion
Lα|δ is the disjoint union Lα ⊔ Lδ, and tα : Lα → Lα|δ, tδ : Lδ → Lα|δ are the obvious injections. For coherence on Lα|δ, (Xα, Xδ ⊢ Yα, Yδ) iff (Xα ⊢ Yα) or (Xδ ⊢ Yδ). This is a coherence relation. The vocabularies slide past one another with no interaction. This ‘translation’ is structure preserving, and coherence and incoherence preserving too.
GregRestall http://consequently.org/presentation/2015/verbal-disputes-aap/ 22 of 61
Resolution by DisjointUnion
Lα|δ is the disjoint union Lα ⊔ Lδ, and tα : Lα → Lα|δ, tδ : Lδ → Lα|δ are the obvious injections. For coherence on Lα|δ, (Xα, Xδ ⊢ Yα, Yδ) iff (Xα ⊢ Yα) or (Xδ ⊢ Yδ). This is a coherence relation. The vocabularies slide past one another with no interaction. This ‘translation’ is structure preserving, and coherence and incoherence preserving too.
GregRestall http://consequently.org/presentation/2015/verbal-disputes-aap/ 22 of 61
This ‘resolves’ the dispute over C If C ̸⊢Lα
( ’s assertion of is coherent)
and
( ’s denial of is coherent)
then
(Asserting
and denying
is coherent.)
GregRestall http://consequently.org/presentation/2015/verbal-disputes-aap/ 23 of 61
This ‘resolves’ the dispute over C If C ̸⊢Lα
(α’s assertion of C is coherent)
and
( ’s denial of is coherent)
then
(Asserting
and denying
is coherent.)
GregRestall http://consequently.org/presentation/2015/verbal-disputes-aap/ 23 of 61
This ‘resolves’ the dispute over C If C ̸⊢Lα
(α’s assertion of C is coherent)
and ̸⊢Lδ C
( ’s denial of is coherent)
then
(Asserting
and denying
is coherent.)
GregRestall http://consequently.org/presentation/2015/verbal-disputes-aap/ 23 of 61
This ‘resolves’ the dispute over C If C ̸⊢Lα
(α’s assertion of C is coherent)
and ̸⊢Lδ C
(δ’s denial of C is coherent)
then
(Asserting
and denying
is coherent.)
GregRestall http://consequently.org/presentation/2015/verbal-disputes-aap/ 23 of 61
This ‘resolves’ the dispute over C If C ̸⊢Lα
(α’s assertion of C is coherent)
and ̸⊢Lδ C
(δ’s denial of C is coherent)
then C ̸⊢Lα|δ C
(Asserting
and denying
is coherent.)
GregRestall http://consequently.org/presentation/2015/verbal-disputes-aap/ 23 of 61
This ‘resolves’ the dispute over C If C ̸⊢Lα
(α’s assertion of C is coherent)
and ̸⊢Lδ C
(δ’s denial of C is coherent)
then C ̸⊢Lα|δ C
(Asserting C-from-Lα and denying C-from-Lδ is coherent.)
GregRestall http://consequently.org/presentation/2015/verbal-disputes-aap/ 23 of 61
Nothing α says has any bearing on δ, or vice versa.
GregRestall http://consequently.org/presentation/2015/verbal-disputes-aap/ 25 of 61
Losing my Conjunction What is A ∧ B? There’s no such sentence in !
GregRestall http://consequently.org/presentation/2015/verbal-disputes-aap/ 26 of 61
Losing my Conjunction What is A ∧ B? There’s no such sentence in Lα|δ!
GregRestall http://consequently.org/presentation/2015/verbal-disputes-aap/ 26 of 61
The Case of the Venusians
Suppose aliens land on earth speaking our languages and familiar with our cultures and tell us that for more complete communication it will be necessary that we increase our vocabulary by the addition of a 1-ary sentence connective V … concerning which we should note immediately that certain restrictions to our familiar inferential practices will need to be
B follows from their conjunction A ∧ B, it will not in general be the case that VA follows from VA ∧ B, or that VB follows from A ∧ VB… — Lloyd Humberstone, The Connectives §4.34
GregRestall http://consequently.org/presentation/2015/verbal-disputes-aap/ 27 of 61
Losing our Conjunction
If some statements A (from Lα) and B (from Lδ) are both deniable (so ̸⊢ A, and ̸⊢ B) then no sentence in Lα|δ entails both A and B. If and then if is in then (possible) and (no). if is in then (possible) and (no). So, there’s no conjunction in .
GregRestall http://consequently.org/presentation/2015/verbal-disputes-aap/ 28 of 61
Losing our Conjunction
If some statements A (from Lα) and B (from Lδ) are both deniable (so ̸⊢ A, and ̸⊢ B) then no sentence in Lα|δ entails both A and B. If C ⊢ A and C ⊢ B then if is in then (possible) and (no). if is in then (possible) and (no). So, there’s no conjunction in .
GregRestall http://consequently.org/presentation/2015/verbal-disputes-aap/ 28 of 61
Losing our Conjunction
If some statements A (from Lα) and B (from Lδ) are both deniable (so ̸⊢ A, and ̸⊢ B) then no sentence in Lα|δ entails both A and B. If C ⊢ A and C ⊢ B then
▶ if C is in Lα then C ⊢ A (possible) and ⊢ B (no).
if is in then (possible) and (no). So, there’s no conjunction in .
GregRestall http://consequently.org/presentation/2015/verbal-disputes-aap/ 28 of 61
Losing our Conjunction
If some statements A (from Lα) and B (from Lδ) are both deniable (so ̸⊢ A, and ̸⊢ B) then no sentence in Lα|δ entails both A and B. If C ⊢ A and C ⊢ B then
▶ if C is in Lα then C ⊢ A (possible) and ⊢ B (no). ▶ if C is in Lδ then C ⊢ B (possible) and ⊢ C (no).
So, there’s no conjunction in .
GregRestall http://consequently.org/presentation/2015/verbal-disputes-aap/ 28 of 61
Losing our Conjunction
If some statements A (from Lα) and B (from Lδ) are both deniable (so ̸⊢ A, and ̸⊢ B) then no sentence in Lα|δ entails both A and B. If C ⊢ A and C ⊢ B then
▶ if C is in Lα then C ⊢ A (possible) and ⊢ B (no). ▶ if C is in Lδ then C ⊢ B (possible) and ⊢ C (no).
So, there’s no conjunction in Lα|δ.
GregRestall http://consequently.org/presentation/2015/verbal-disputes-aap/ 28 of 61
Have we got conjunction in L?
We can mean many different things by ‘and’. Let’s say that ‘and’ is a conjunction in iff: and and for all , , and in .
(There is no conjunction in . There is no sentence “ and ”.)
GregRestall http://consequently.org/presentation/2015/verbal-disputes-aap/ 30 of 61
Have we got conjunction in L?
We can mean many different things by ‘and’. Let’s say that ‘and’ is a conjunction in iff: and and for all , , and in .
(There is no conjunction in . There is no sentence “ and ”.)
GregRestall http://consequently.org/presentation/2015/verbal-disputes-aap/ 30 of 61
Have we got conjunction in L?
We can mean many different things by ‘and’. Let’s say that ‘and’ is a conjunction in L iff: and and for all , , and in .
(There is no conjunction in . There is no sentence “ and ”.)
GregRestall http://consequently.org/presentation/2015/verbal-disputes-aap/ 30 of 61
Have we got conjunction in L?
We can mean many different things by ‘and’. Let’s say that ‘and’ is a conjunction in L iff: X, A, B ⊢ Y = = = = = = = = = = = = [and↕] X, A and B ⊢ Y for all X, Y, A and B in L.
(There is no conjunction in . There is no sentence “ and ”.)
GregRestall http://consequently.org/presentation/2015/verbal-disputes-aap/ 30 of 61
Have we got conjunction in L?
We can mean many different things by ‘and’. Let’s say that ‘and’ is a conjunction in L iff: X, A, B ⊢ Y = = = = = = = = = = = = [and↕] X, A and B ⊢ Y for all X, Y, A and B in L.
(There is no conjunction in Lα|δ. There is no sentence “A and B”.)
GregRestall http://consequently.org/presentation/2015/verbal-disputes-aap/ 30 of 61
Preservation A translation t : L1 → L2 is conjunction preserving if a conjunction in L1 is translated by a conjunction in L2.
GregRestall http://consequently.org/presentation/2015/verbal-disputes-aap/ 31 of 61
Preservation seems like a good idea Translations should keep some things preserved. Let’s see what we can do with this.
GregRestall http://consequently.org/presentation/2015/verbal-disputes-aap/ 32 of 61
Conjunction
Obviously, there some disagreements can resolved by a disambiguation of different senses of the word ‘and.’ ‘and ’ ‘ ’ ‘and ’ ‘and then’
GregRestall http://consequently.org/presentation/2015/verbal-disputes-aap/ 34 of 61
Conjunction
Obviously, there some disagreements can resolved by a disambiguation of different senses of the word ‘and.’ ‘andα’
tα
− → ‘∧’ ‘andδ’
tδ
− → ‘and then’
GregRestall http://consequently.org/presentation/2015/verbal-disputes-aap/ 34 of 61
No Verbal Disagreement Between Two Conjunctions
If the following two conditions hold:
and ‘ ’ is a conjunction in , and 2. , and are both conjunction preserving. then ‘ ’ and ‘ ’ are equivalent in . That is, in , and . Why?
GregRestall http://consequently.org/presentation/2015/verbal-disputes-aap/ 35 of 61
No Verbal Disagreement Between Two Conjunctions
If the following two conditions hold:
2. , and are both conjunction preserving. then ‘ ’ and ‘ ’ are equivalent in . That is, in , and . Why?
GregRestall http://consequently.org/presentation/2015/verbal-disputes-aap/ 35 of 61
No Verbal Disagreement Between Two Conjunctions
If the following two conditions hold:
then ‘ ’ and ‘ ’ are equivalent in . That is, in , and . Why?
GregRestall http://consequently.org/presentation/2015/verbal-disputes-aap/ 35 of 61
No Verbal Disagreement Between Two Conjunctions
If the following two conditions hold:
then ‘∧’ and ‘&’ are equivalent in L∗. That is, in , and . Why?
GregRestall http://consequently.org/presentation/2015/verbal-disputes-aap/ 35 of 61
No Verbal Disagreement Between Two Conjunctions
If the following two conditions hold:
then ‘∧’ and ‘&’ are equivalent in L∗. That is, in L∗, A ∧ B ⊢ A & B and A & B ⊢ A ∧ B. Why?
GregRestall http://consequently.org/presentation/2015/verbal-disputes-aap/ 35 of 61
No Verbal Disagreement Between Two Conjunctions
If the following two conditions hold:
then ‘∧’ and ‘&’ are equivalent in L∗. That is, in L∗, A ∧ B ⊢ A & B and A & B ⊢ A ∧ B. Why?
GregRestall http://consequently.org/presentation/2015/verbal-disputes-aap/ 35 of 61
Here's why
Reason as follows inside L∗: (Since and are both conjunctions in .)
GregRestall http://consequently.org/presentation/2015/verbal-disputes-aap/ 36 of 61
Here's why
Reason as follows inside L∗: A & B ⊢ A & B [&↑] A, B ⊢ A & B [∧↓] A ∧ B ⊢ A & B A ∧ B ⊢ A ∧ B [∧↑] A, B ⊢ A ∧ B [&↓] A & B ⊢ A ∧ B (Since ∧ and & are both conjunctions in L∗.)
GregRestall http://consequently.org/presentation/2015/verbal-disputes-aap/ 36 of 61
Equivalence and Verbal Disagreements
If ‘∧’ and ‘&’ are equivalent, then any merely verbal disagreement betwen A ∧ B and A′&B′ cannot be explained by an equivocation between ‘∧’ and ‘&’. The only way to coherently assert and deny involves distinguishing and
and . Cut Cut If / and / are equivalent, so are and .
GregRestall http://consequently.org/presentation/2015/verbal-disputes-aap/ 37 of 61
Equivalence and Verbal Disagreements
If ‘∧’ and ‘&’ are equivalent, then any merely verbal disagreement betwen A ∧ B and A′&B′ cannot be explained by an equivocation between ‘∧’ and ‘&’. The only way to coherently assert A ∧ B and deny A′ & B′ involves distinguishing A and A′ or B and B′. Cut Cut If / and / are equivalent, so are and .
GregRestall http://consequently.org/presentation/2015/verbal-disputes-aap/ 37 of 61
Equivalence and Verbal Disagreements
If ‘∧’ and ‘&’ are equivalent, then any merely verbal disagreement betwen A ∧ B and A′&B′ cannot be explained by an equivocation between ‘∧’ and ‘&’. The only way to coherently assert A ∧ B and deny A′ & B′ involves distinguishing A and A′ or B and B′. A ⊢ A′ B ⊢ B′ A′ & B′ ⊢ A′ & B′ [&↑] A′, B′ ⊢ A′ & B′ [Cut] A′, B ⊢ A′ & B′ [Cut] A, B ⊢ A′ & B′ [∧↓] A ∧ B ⊢ A′ & B′ If A/A′ and B/B′are equivalent, so are A ∧ B and A′ & B′.
GregRestall http://consequently.org/presentation/2015/verbal-disputes-aap/ 37 of 61
This is not surprising… … since the rules for conjunction are very strong.
GregRestall http://consequently.org/presentation/2015/verbal-disputes-aap/ 38 of 61
This is not surprising… … since the rules for conjunction are very strong.
GregRestall http://consequently.org/presentation/2015/verbal-disputes-aap/ 38 of 61
Negation
Consider the debate between the intuitionist and classical logician over negation. Dummett: I assert and deny : . Williamson: . Could this be a merely verbal disagreement? Of course! There are logics in which both intuitionist and classical ‘negation’ can be distinguished. Sort of.
GregRestall http://consequently.org/presentation/2015/verbal-disputes-aap/ 39 of 61
Negation
Consider the debate between the intuitionist and classical logician over negation. Dummett: I assert ¬¬p and deny p: ¬¬p ̸⊢ p. Williamson: . Could this be a merely verbal disagreement? Of course! There are logics in which both intuitionist and classical ‘negation’ can be distinguished. Sort of.
GregRestall http://consequently.org/presentation/2015/verbal-disputes-aap/ 39 of 61
Negation
Consider the debate between the intuitionist and classical logician over negation. Dummett: I assert ¬¬p and deny p: ¬¬p ̸⊢ p. Williamson: −−p ⊢ p. Could this be a merely verbal disagreement? Of course! There are logics in which both intuitionist and classical ‘negation’ can be distinguished. Sort of.
GregRestall http://consequently.org/presentation/2015/verbal-disputes-aap/ 39 of 61
Negation
Consider the debate between the intuitionist and classical logician over negation. Dummett: I assert ¬¬p and deny p: ¬¬p ̸⊢ p. Williamson: −−p ⊢ p. Could this be a merely verbal disagreement? Of course! There are logics in which both intuitionist and classical ‘negation’ can be distinguished. Sort of.
GregRestall http://consequently.org/presentation/2015/verbal-disputes-aap/ 39 of 61
Negation
Consider the debate between the intuitionist and classical logician over negation. Dummett: I assert ¬¬p and deny p: ¬¬p ̸⊢ p. Williamson: −−p ⊢ p. Could this be a merely verbal disagreement? Of course! There are logics in which both intuitionist and classical ‘negation’ can be distinguished. Sort of.
GregRestall http://consequently.org/presentation/2015/verbal-disputes-aap/ 39 of 61
Negation
Consider the debate between the intuitionist and classical logician over negation. Dummett: I assert ¬¬p and deny p: ¬¬p ̸⊢ p. Williamson: −−p ⊢ p. Could this be a merely verbal disagreement? Of course! There are logics in which both intuitionist and classical ‘negation’ can be distinguished. Sort of.
GregRestall http://consequently.org/presentation/2015/verbal-disputes-aap/ 39 of 61
Negation
When is something a negation? classical logic: intuitionist logic: Let’s call something a negation in if it satisfies at least the intuitionist negation rules. And let’s say that preserves negation if it translates a negation in by a negation in .
GregRestall http://consequently.org/presentation/2015/verbal-disputes-aap/ 40 of 61
Negation
When is something a negation? classical logic: X ⊢ A, Y = = = = = = = = [−↕] X, −A ⊢ Y intuitionist logic: Let’s call something a negation in if it satisfies at least the intuitionist negation rules. And let’s say that preserves negation if it translates a negation in by a negation in .
GregRestall http://consequently.org/presentation/2015/verbal-disputes-aap/ 40 of 61
Negation
When is something a negation? classical logic: X ⊢ A, Y = = = = = = = = [−↕] X, −A ⊢ Y intuitionist logic: X, A ⊢ = = = = = = [¬↕] X ⊢ ¬A Let’s call something a negation in if it satisfies at least the intuitionist negation rules. And let’s say that preserves negation if it translates a negation in by a negation in .
GregRestall http://consequently.org/presentation/2015/verbal-disputes-aap/ 40 of 61
Negation
When is something a negation? classical logic: X ⊢ A, Y = = = = = = = = [−↕] X, −A ⊢ Y intuitionist logic: X, A ⊢ = = = = = = [¬↕] X ⊢ ¬A Let’s call something a negation in L if it satisfies at least the intuitionist negation rules. And let’s say that preserves negation if it translates a negation in by a negation in .
GregRestall http://consequently.org/presentation/2015/verbal-disputes-aap/ 40 of 61
Negation
When is something a negation? classical logic: X ⊢ A, Y = = = = = = = = [−↕] X, −A ⊢ Y intuitionist logic: X, A ⊢ = = = = = = [¬↕] X ⊢ ¬A Let’s call something a negation in L if it satisfies at least the intuitionist negation rules. And let’s say that t : L1 → L2 preserves negation if it translates a negation in L1 by a negation in L2.
GregRestall http://consequently.org/presentation/2015/verbal-disputes-aap/ 40 of 61
No Verbal Disagreement Between Two Negations
If the following two conditions hold:
and ‘ ’ is a negation in , and 2. , and are both negation preserving. then ‘ ’ and ‘ ’ are equivalent in . That is, in , and . Why?
GregRestall http://consequently.org/presentation/2015/verbal-disputes-aap/ 41 of 61
No Verbal Disagreement Between Two Negations
If the following two conditions hold:
2. , and are both negation preserving. then ‘ ’ and ‘ ’ are equivalent in . That is, in , and . Why?
GregRestall http://consequently.org/presentation/2015/verbal-disputes-aap/ 41 of 61
No Verbal Disagreement Between Two Negations
If the following two conditions hold:
then ‘ ’ and ‘ ’ are equivalent in . That is, in , and . Why?
GregRestall http://consequently.org/presentation/2015/verbal-disputes-aap/ 41 of 61
No Verbal Disagreement Between Two Negations
If the following two conditions hold:
then ‘¬’ and ‘−’ are equivalent in L∗. That is, in , and . Why?
GregRestall http://consequently.org/presentation/2015/verbal-disputes-aap/ 41 of 61
No Verbal Disagreement Between Two Negations
If the following two conditions hold:
then ‘¬’ and ‘−’ are equivalent in L∗. That is, in L∗, ¬A ⊢ −A and −A ⊢ ¬A. Why?
GregRestall http://consequently.org/presentation/2015/verbal-disputes-aap/ 41 of 61
No Verbal Disagreement Between Two Negations
If the following two conditions hold:
then ‘¬’ and ‘−’ are equivalent in L∗. That is, in L∗, ¬A ⊢ −A and −A ⊢ ¬A. Why?
GregRestall http://consequently.org/presentation/2015/verbal-disputes-aap/ 41 of 61
Collapse?
Reason as follows inside L∗: −A ⊢ −A [−↑] −A, A ⊢ [¬↓] −A ⊢ ¬A ¬A ⊢ ¬A [¬↑] ¬A, A ⊢ [−↓] ¬A ⊢ −A It follows that any disagreement, where one asserts ¬A and the other denies −A (or vice versa) must resolve into a disagreement over A.
GregRestall http://consequently.org/presentation/2015/verbal-disputes-aap/ 42 of 61
Equivalence and Verbal Disagreements: The Negation Case
If ‘¬’ and ‘−’ are equivalent, then any merrely verbal disagreement between ¬A and −A′ cannot be explained by an equivocation between the two negations. The only way to coherently assert ¬A and deny −A′ involves distinguishing A and A′. ¬A ⊢ ¬A [¬↑] ¬A, A ⊢ A ⊢ A′ [Cut] ¬A, A′ ⊢ [−↓] ¬A ⊢ −A′
GregRestall http://consequently.org/presentation/2015/verbal-disputes-aap/ 43 of 61
What options are there for disagreement?
▶ Disagreement over the consequence relation ‘⊢’ (pluralism). ▶ The classical logician thinks the intuitionist is mistaken to take ‘¬’
to be so weak, or the intuitionist thinks that the classical logician is mistaken to take ‘−’ to be so strong.
GregRestall http://consequently.org/presentation/2015/verbal-disputes-aap/ 44 of 61
Ontological Relativity
Can we have merely verbal disagreement about ‘exists’? Can we have merely verbal disagreement about ‘ ’? Surely! Take multi-sorted first order logic. says that there are numbers ( ). denies it ( ). Can we make this difference merely verbal? While respecting some of the semantics of ?
Translate into a vocabulary with two quantifiers and two two domains: and with two quantifiers and ranging over each. Let have a non-empty extension on but an empty one on . Both and can happily endorse and deny and be done with it. Isn’t this a merely verbal disagreement over what exists?
GregRestall http://consequently.org/presentation/2015/verbal-disputes-aap/ 45 of 61
Ontological Relativity
Can we have merely verbal disagreement about ‘exists’? Can we have merely verbal disagreement about ‘(∃x)’? Surely! Take multi-sorted first order logic. says that there are numbers ( ). denies it ( ). Can we make this difference merely verbal? While respecting some of the semantics of ?
Translate into a vocabulary with two quantifiers and two two domains: and with two quantifiers and ranging over each. Let have a non-empty extension on but an empty one on . Both and can happily endorse and deny and be done with it. Isn’t this a merely verbal disagreement over what exists?
GregRestall http://consequently.org/presentation/2015/verbal-disputes-aap/ 45 of 61
Ontological Relativity
Can we have merely verbal disagreement about ‘exists’? Can we have merely verbal disagreement about ‘(∃x)’? Surely! Take multi-sorted first order logic. says that there are numbers ( ). denies it ( ). Can we make this difference merely verbal? While respecting some of the semantics of ?
Translate into a vocabulary with two quantifiers and two two domains: and with two quantifiers and ranging over each. Let have a non-empty extension on but an empty one on . Both and can happily endorse and deny and be done with it. Isn’t this a merely verbal disagreement over what exists?
GregRestall http://consequently.org/presentation/2015/verbal-disputes-aap/ 45 of 61
Ontological Relativity
Can we have merely verbal disagreement about ‘exists’? Can we have merely verbal disagreement about ‘(∃x)’? Surely! Take multi-sorted first order logic. α says that there are numbers ((∃x)Nx). δ denies it (¬(∃x)Nx). Can we make this difference merely verbal? While respecting some of the semantics of (∃x)?
Translate into a vocabulary with two quantifiers and two two domains: and with two quantifiers and ranging over each. Let have a non-empty extension on but an empty one on . Both and can happily endorse and deny and be done with it. Isn’t this a merely verbal disagreement over what exists?
GregRestall http://consequently.org/presentation/2015/verbal-disputes-aap/ 45 of 61
Ontological Relativity
Can we have merely verbal disagreement about ‘exists’? Can we have merely verbal disagreement about ‘(∃x)’? Surely! Take multi-sorted first order logic. α says that there are numbers ((∃x)Nx). δ denies it (¬(∃x)Nx). Can we make this difference merely verbal? While respecting some of the semantics of (∃x)?
Translate into a vocabulary with two quantifiers and two two domains: D1 and D2 with two quantifiers (∃1x) and (∃2x) ranging over each. Let N have a non-empty extension on D1 but an empty one on D2. Both α and δ can happily endorse (∃1x)Nx and deny (∃2x)Nx and be done with it. Isn’t this a merely verbal disagreement over what exists?
GregRestall http://consequently.org/presentation/2015/verbal-disputes-aap/ 45 of 61
Ontological Relativity
Can we have merely verbal disagreement about ‘exists’? Can we have merely verbal disagreement about ‘(∃x)’? Surely! Take multi-sorted first order logic. α says that there are numbers ((∃x)Nx). δ denies it (¬(∃x)Nx). Can we make this difference merely verbal? While respecting some of the semantics of (∃x)?
Translate into a vocabulary with two quantifiers and two two domains: D1 and D2 with two quantifiers (∃1x) and (∃2x) ranging over each. Let N have a non-empty extension on D1 but an empty one on D2. Both α and δ can happily endorse (∃1x)Nx and deny (∃2x)Nx and be done with it. Isn’t this a merely verbal disagreement over what exists?
GregRestall http://consequently.org/presentation/2015/verbal-disputes-aap/ 45 of 61
Not so fast…
Perhaps there is scope for the same behaviour as with conjunction and
existential quantifier, and a translation preserving it. (Where is not free in and .) This is what it takes to be an existential quantifier in .
GregRestall http://consequently.org/presentation/2015/verbal-disputes-aap/ 46 of 61
Not so fast…
Perhaps there is scope for the same behaviour as with conjunction and negation. Consider more closely what might be involved in being an existential quantifier, and a translation preserving it. (Where is not free in and .) This is what it takes to be an existential quantifier in .
GregRestall http://consequently.org/presentation/2015/verbal-disputes-aap/ 46 of 61
Not so fast…
Perhaps there is scope for the same behaviour as with conjunction and
existential quantifier, and a translation preserving it. (Where is not free in and .) This is what it takes to be an existential quantifier in .
GregRestall http://consequently.org/presentation/2015/verbal-disputes-aap/ 46 of 61
Not so fast…
Perhaps there is scope for the same behaviour as with conjunction and
existential quantifier, and a translation preserving it. X, A(v) ⊢ Y = = = = = = = = = = = = = [∃↕] X, (∃x)A(x) ⊢ Y (Where v is not free in X and Y.) This is what it takes to be an existential quantifier in L.
GregRestall http://consequently.org/presentation/2015/verbal-disputes-aap/ 46 of 61
Existential Quantifier Collapse
(∃2x)A(x) ⊢ (∃2x)A(x) [∃2↑] A(v) ⊢ (∃2x)A(x) [∃1↓] (∃1x)A(x) ⊢ (∃2x)A(x) (∃1x)A(x) ⊢ (∃1x)A(x) [∃1↑] A(v) ⊢ (∃1x)A(x) [∃2↓] (∃2x)A(x) ⊢ (∃1x)A(x) If the term appropriate to also applies in , and vice versa, then indeed, the two quantifiers collapse.
GregRestall http://consequently.org/presentation/2015/verbal-disputes-aap/ 47 of 61
Existential Quantifier Collapse
(∃2x)A(x) ⊢ (∃2x)A(x) [∃2↑] A(v) ⊢ (∃2x)A(x) [∃1↓] (∃1x)A(x) ⊢ (∃2x)A(x) (∃1x)A(x) ⊢ (∃1x)A(x) [∃1↑] A(v) ⊢ (∃1x)A(x) [∃2↓] (∃2x)A(x) ⊢ (∃1x)A(x) If the term v appropriate to [∃1↕] also applies in [∃2↕], and vice versa, then indeed, the two quantifiers collapse.
GregRestall http://consequently.org/presentation/2015/verbal-disputes-aap/ 47 of 61
Coordination on terms brings coordination on (∃x)
If the following three conditions hold:
quantifier in L2, and
and
then (∃1x) and (∃2x) are equivalent in L∗, in that in L∗ we have (∃1x)A ⊢ (∃2x)A and (∃2x)A ⊢ (∃1x)A.
GregRestall http://consequently.org/presentation/2015/verbal-disputes-aap/ 48 of 61
Coordination on terms brings coordination on (∃x)
If the following three conditions hold:
quantifier in L2, and
and
then (∃1x) and (∃2x) are equivalent in L∗, in that in L∗ we have (∃1x)A ⊢ (∃2x)A and (∃2x)A ⊢ (∃1x)A.
GregRestall http://consequently.org/presentation/2015/verbal-disputes-aap/ 48 of 61
It's important to recognise what this is not
The appropriateness condition for eigenvariables (demonstratives, terms) is
You could coherently be a monist and argue with someone with a more conventional ontology—with the same quantifiers—provided that you both took the same terms (demonstratives, eigenvariables, whatever) to be in order for that quantifier. You don’t need to take these terms to refer to (or range over) the same things in any substantial sense.
GregRestall http://consequently.org/presentation/2015/verbal-disputes-aap/ 49 of 61
It's important to recognise what this is not
The appropriateness condition for eigenvariables (demonstratives, terms) is
You could coherently be a monist and argue with someone with a more conventional ontology—with the same quantifiers—provided that you both took the same terms (demonstratives, eigenvariables, whatever) to be in order for that quantifier. You don’t need to take these terms to refer to (or range over) the same things in any substantial sense.
GregRestall http://consequently.org/presentation/2015/verbal-disputes-aap/ 49 of 61
It's important to recognise what this is not
The appropriateness condition for eigenvariables (demonstratives, terms) is
You could coherently be a monist and argue with someone with a more conventional ontology—with the same quantifiers—provided that you both took the same terms (demonstratives, eigenvariables, whatever) to be in order for that quantifier. You don’t need to take these terms to refer to (or range over) the same things in any substantial sense.
GregRestall http://consequently.org/presentation/2015/verbal-disputes-aap/ 49 of 61
A Monist arguing with a Pluralist (agreeing on terms)
monist:
▶ (∀x)(∀y)x = y
, pluralist: ,
GregRestall http://consequently.org/presentation/2015/verbal-disputes-aap/ 50 of 61
A Monist arguing with a Pluralist (agreeing on terms)
monist:
▶ (∀x)(∀y)x = y
, pluralist:
▶ (∃x)(∃y)x ̸= y
,
GregRestall http://consequently.org/presentation/2015/verbal-disputes-aap/ 50 of 61
A Monist arguing with a Pluralist (agreeing on terms)
monist:
▶ (∀x)(∀y)x = y
, pluralist:
▶ (∃x)(∃y)x ̸= y ▶ (∃y)a ̸= y
,
GregRestall http://consequently.org/presentation/2015/verbal-disputes-aap/ 50 of 61
A Monist arguing with a Pluralist (agreeing on terms)
monist:
▶ (∀x)(∀y)x = y ▶ (∀y)a = y
, pluralist:
▶ (∃x)(∃y)x ̸= y ▶ (∃y)a ̸= y
,
GregRestall http://consequently.org/presentation/2015/verbal-disputes-aap/ 50 of 61
A Monist arguing with a Pluralist (agreeing on terms)
monist:
▶ (∀x)(∀y)x = y ▶ (∀y)a = y
, pluralist:
▶ (∃x)(∃y)x ̸= y ▶ (∃y)a ̸= y ▶ a ̸= b
,
GregRestall http://consequently.org/presentation/2015/verbal-disputes-aap/ 50 of 61
A Monist arguing with a Pluralist (agreeing on terms)
monist:
▶ (∀x)(∀y)x = y ▶ (∀y)a = y ▶ a = b
, pluralist:
▶ (∃x)(∃y)x ̸= y ▶ (∃y)a ̸= y ▶ a ̸= b
,
GregRestall http://consequently.org/presentation/2015/verbal-disputes-aap/ 50 of 61
A Monist arguing with a Pluralist (agreeing on terms)
monist:
▶ (∀x)(∀y)x = y ▶ (∀y)a = y ▶ a = b
, pluralist:
▶ (∃x)(∃y)x ̸= y ▶ (∃y)a ̸= y ▶ a ̸= b ▶ Fa, ¬Fb
GregRestall http://consequently.org/presentation/2015/verbal-disputes-aap/ 50 of 61
A Monist arguing with a Pluralist (agreeing on terms)
monist:
▶ (∀x)(∀y)x = y ▶ (∀y)a = y ▶ a = b ▶ Fa, Fb
pluralist:
▶ (∃x)(∃y)x ̸= y ▶ (∃y)a ̸= y ▶ a ̸= b ▶ Fa, ¬Fb
GregRestall http://consequently.org/presentation/2015/verbal-disputes-aap/ 50 of 61
A Monist arguing with a Pluralist (agreeing on terms)
monist:
▶ (∀x)(∀y)x = y ▶ (∀y)a = y ▶ a = b ▶ Fa, Fb
pluralist:
▶ (∃x)(∃y)x ̸= y ▶ (∃y)a ̸= y ▶ a ̸= b ▶ Fa, ¬Fb
GregRestall http://consequently.org/presentation/2015/verbal-disputes-aap/ 50 of 61
A Monist arguing with a Pluralist (disagreeing on terms)
If the pluralist had argued instead:
▶ (∃x)(∃y)x ̸= y, because
, since is commutative and is not, It’s fair for the monist (or anyone else) to agree is commutative, and is not But to not take these to be predications of the form and , and so, to not be appropriate to substitute into the quantifier.
GregRestall http://consequently.org/presentation/2015/verbal-disputes-aap/ 51 of 61
A Monist arguing with a Pluralist (disagreeing on terms)
If the pluralist had argued instead:
▶ (∃x)(∃y)x ̸= y, because ▶ ∧ ̸= ⊃, since
is commutative and is not, It’s fair for the monist (or anyone else) to agree is commutative, and is not But to not take these to be predications of the form and , and so, to not be appropriate to substitute into the quantifier.
GregRestall http://consequently.org/presentation/2015/verbal-disputes-aap/ 51 of 61
A Monist arguing with a Pluralist (disagreeing on terms)
If the pluralist had argued instead:
▶ (∃x)(∃y)x ̸= y, because ▶ ∧ ̸= ⊃, since ▶ ∧ is commutative and ⊃ is not,
It’s fair for the monist (or anyone else) to agree is commutative, and is not But to not take these to be predications of the form and , and so, to not be appropriate to substitute into the quantifier.
GregRestall http://consequently.org/presentation/2015/verbal-disputes-aap/ 51 of 61
A Monist arguing with a Pluralist (disagreeing on terms)
If the pluralist had argued instead:
▶ (∃x)(∃y)x ̸= y, because ▶ ∧ ̸= ⊃, since ▶ ∧ is commutative and ⊃ is not,
It’s fair for the monist (or anyone else) to agree is commutative, and is not But to not take these to be predications of the form and , and so, to not be appropriate to substitute into the quantifier.
GregRestall http://consequently.org/presentation/2015/verbal-disputes-aap/ 51 of 61
A Monist arguing with a Pluralist (disagreeing on terms)
If the pluralist had argued instead:
▶ (∃x)(∃y)x ̸= y, because ▶ ∧ ̸= ⊃, since ▶ ∧ is commutative and ⊃ is not,
It’s fair for the monist (or anyone else) to agree
▶ ∧ is commutative, and ⊃ is not
But to not take these to be predications of the form and , and so, to not be appropriate to substitute into the quantifier.
GregRestall http://consequently.org/presentation/2015/verbal-disputes-aap/ 51 of 61
A Monist arguing with a Pluralist (disagreeing on terms)
If the pluralist had argued instead:
▶ (∃x)(∃y)x ̸= y, because ▶ ∧ ̸= ⊃, since ▶ ∧ is commutative and ⊃ is not,
It’s fair for the monist (or anyone else) to agree
▶ ∧ is commutative, and ⊃ is not
But to not take these to be predications of the form Fa and ¬Fb, and so, to not be appropriate to substitute into the quantifier.
GregRestall http://consequently.org/presentation/2015/verbal-disputes-aap/ 51 of 61
Modal Relativity
Can we have merely verbal disagreement about ‘possibility’? Can we have merely verbal disagreement about ‘ ’? Surely! Take multi-modal logic. ranges over possible worlds; ranges over times. Isn’t this a merely verbal disagreement over what possible?
GregRestall http://consequently.org/presentation/2015/verbal-disputes-aap/ 52 of 61
Modal Relativity
Can we have merely verbal disagreement about ‘possibility’? Can we have merely verbal disagreement about ‘♢’? Surely! Take multi-modal logic. ranges over possible worlds; ranges over times. Isn’t this a merely verbal disagreement over what possible?
GregRestall http://consequently.org/presentation/2015/verbal-disputes-aap/ 52 of 61
Modal Relativity
Can we have merely verbal disagreement about ‘possibility’? Can we have merely verbal disagreement about ‘♢’? Surely! Take multi-modal logic. ranges over possible worlds; ranges over times. Isn’t this a merely verbal disagreement over what possible?
GregRestall http://consequently.org/presentation/2015/verbal-disputes-aap/ 52 of 61
Modal Relativity
Can we have merely verbal disagreement about ‘possibility’? Can we have merely verbal disagreement about ‘♢’? Surely! Take multi-modal logic. ♢1 ranges over possible worlds; ♢2 ranges over times. Isn’t this a merely verbal disagreement over what possible?
GregRestall http://consequently.org/presentation/2015/verbal-disputes-aap/ 52 of 61
Modal Relativity
Can we have merely verbal disagreement about ‘possibility’? Can we have merely verbal disagreement about ‘♢’? Surely! Take multi-modal logic. ♢1 ranges over possible worlds; ♢2 ranges over times. Isn’t this a merely verbal disagreement over what possible?
GregRestall http://consequently.org/presentation/2015/verbal-disputes-aap/ 52 of 61
Not so fast…
Let’s consider more closely what might be involved in possibility preservation. A ⊢ | X ⊢ Y | ∆ = = = = = = = = = = = = [♢↕] X, ♢A ⊢ Y | ∆ The separated sequents indicate positions in which assertions and denials are made in different zones of a discourse.
For details, see Greg Restall “Proofnets for S5” pages 151–172 in Logic Colloquium 2005, C. Dimitracopoulos,
Press, 2007 «http://consequently.org/writing/s5nets/» Greg Restall “A Cut-Free Sequent System for Two-Dimensional Modal Logic—and why it matters,” Annals of Pure and Applied Logic 2012 (163) 1611–1623. «http://consequently.org/writing/cfss2dml/»
GregRestall http://consequently.org/presentation/2015/verbal-disputes-aap/ 53 of 61
Not so fast…
Let’s consider more closely what might be involved in possibility preservation. A ⊢ | X ⊢ Y | ∆ = = = = = = = = = = = = [♢↕] X, ♢A ⊢ Y | ∆ The separated sequents indicate positions in which assertions and denials are made in different zones of a discourse.
For details, see
▶ Greg Restall “Proofnets for S5” pages 151–172 in Logic Colloquium 2005, C. Dimitracopoulos,
Press, 2007 «http://consequently.org/writing/s5nets/»
▶ Greg Restall “A Cut-Free Sequent System for Two-Dimensional Modal Logic—and why it
matters,” Annals of Pure and Applied Logic 2012 (163) 1611–1623. «http://consequently.org/writing/cfss2dml/»
GregRestall http://consequently.org/presentation/2015/verbal-disputes-aap/ 53 of 61
Possibility
♢2A ⊢ ♢2A [♢2↑] A ⊢ | ⊢ ♢2A [♢1↓] ♢1A ⊢ ♢2A ♢1A ⊢ ♢1A [♢1↑] A ⊢ | ⊢ ♢1A [♢2↓] ♢2A ⊢ ♢1A If the zone appropriate to [♢1↕] also applies in [♢2↕], and vice versa then indeed, the two operators collapse.
GregRestall http://consequently.org/presentation/2015/verbal-disputes-aap/ 54 of 61
Coordination on zones brings coordination on ♢
If the following three conditions hold:
then ♢1 and ♢2 are equivalent in L∗, in that in L∗ we have ♢1A ⊢ ♢2A and ♢2A ⊢ ♢1A.
GregRestall http://consequently.org/presentation/2015/verbal-disputes-aap/ 55 of 61
Coordination on zones brings coordination on ♢
If the following three conditions hold:
then ♢1 and ♢2 are equivalent in L∗, in that in L∗ we have ♢1A ⊢ ♢2A and ♢2A ⊢ ♢1A.
GregRestall http://consequently.org/presentation/2015/verbal-disputes-aap/ 55 of 61
It's important to recognise what this is not
The appropriateness condition for zones is dialogical. It doesn’t force agreement on what is possible. You could coherently be a modal fatalist and argue with someone with a more conventional modal views—with the same modal operators, provided that you both took the same zones to be in order. (You don’t need to take the same things to hold in each zone.)
GregRestall http://consequently.org/presentation/2015/verbal-disputes-aap/ 56 of 61
It's important to recognise what this is not
The appropriateness condition for zones is dialogical. It doesn’t force agreement on what is possible. You could coherently be a modal fatalist and argue with someone with a more conventional modal views—with the same modal operators, provided that you both took the same zones to be in order. (You don’t need to take the same things to hold in each zone.)
GregRestall http://consequently.org/presentation/2015/verbal-disputes-aap/ 56 of 61
It's important to recognise what this is not
The appropriateness condition for zones is dialogical. It doesn’t force agreement on what is possible. You could coherently be a modal fatalist and argue with someone with a more conventional modal views—with the same modal operators, provided that you both took the same zones to be in order. (You don’t need to take the same things to hold in each zone.)
GregRestall http://consequently.org/presentation/2015/verbal-disputes-aap/ 56 of 61
Upshot #1: The Power of Keeping Some Things Fixed The more you want from a translation, the fewer translations you have, and the fewer ways there are to settle disputes as merely verbal. And the more chance you have to locate that dispute in some particular part of your vocabulary.
GregRestall http://consequently.org/presentation/2015/verbal-disputes-aap/ 58 of 61
Upshot #1: The Power of Keeping Some Things Fixed The more you want from a translation, the fewer translations you have, and the fewer ways there are to settle disputes as merely verbal. And the more chance you have to locate that dispute in some particular part of your vocabulary.
GregRestall http://consequently.org/presentation/2015/verbal-disputes-aap/ 58 of 61
Upshot #2: Defining Rules Provide Fixed Points It’s one thing to think of a logical concept as something satisfying a set of axioms. But that is cheap. Defining rules are more powerful.
And defining rules are natural, given the conception of logical constants as topic neutral, and definable in general terms.
GregRestall http://consequently.org/presentation/2015/verbal-disputes-aap/ 59 of 61
Upshot #2: Defining Rules Provide Fixed Points It’s one thing to think of a logical concept as something satisfying a set of axioms. But that is cheap. Defining rules are more powerful.
And defining rules are natural, given the conception of logical constants as topic neutral, and definable in general terms.
GregRestall http://consequently.org/presentation/2015/verbal-disputes-aap/ 59 of 61
Upshot #3: Generality Comes in Degrees
Using this structure to define the behaviour of a logical concepts allows for them to be preserved in translation and used as a fixed point in the midst of disagreement.
GregRestall http://consequently.org/presentation/2015/verbal-disputes-aap/ 60 of 61
Upshot #3: Generality Comes in Degrees
Using this structure to define the behaviour of a logical concepts allows for them to be preserved in translation and used as a fixed point in the midst of disagreement.
GregRestall http://consequently.org/presentation/2015/verbal-disputes-aap/ 60 of 61
http://consequently.org/presentation/2015/verbal-disputes-aap/ @consequently on Twitter